# Questions tagged [groupoids]

A groupoid (in the sense of category theory) is a small category in which every morphism is an isomorphism. Groupoids arise throughout mathematics, e.g. in guise of fundamental groupoids in the theory of covering spaces, holonomy groupoids in the theory of foliations or Lie groupoids in mathematical physics. For groupoids in the sense of universal algebra, i.e., a set with a binary operation, please use the (magma) tag.

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### Inversion on an internal groupoid

Let $\mathcal{C}$ be a category with pullbacks, with $\mathscr{G}=({\bf Ob}_\mathscr{G},{\bf Hom}_\mathscr{G},cod,dom,{\bf 1}, \circ_{\mathscr{G}},-^{-1})$ an internal groupoid in $\mathcal{C}$. I'm ...
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### Faithful representation of $C_c(X \times X)$.

Let $X$ be a smooth manifold. $X \times X$ is product manifold. $\mu$ is a Borel measure on $X$. There are two aims (i) Associate a $C^*$ norm to $C_c(X\times X)$, making it a $C^*$ algebra. (ii)...
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### Adjoint for pullback of sheaves on a topological groupoid

Consider a topological groupoid $G\xrightarrow{s} X$, with the arrow representing the domain map that associates to each morphism its domain. This induces a pullback functor $s^*:Sh(X)\to Sh(G)$. I ...
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### “Eilenberg-MacLane property” for the classifying space of a groupoid

Given a groupoid $G$, its classifying space is defined as the standard geometric realisation of the nerve. My question is: since the classifying space of a group is the only space up to homotopy that ...
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### All sheaves in a Grothendieck topos are generated by subobjects of powers of a suitable sheaf

Consider a topos $\mathcal E$. Butz and Moerdijk, in Representing topoi by topological groupoids, par. 2, say that one can find an object $S\in \mathcal E$ such that the subobjects of its powers (i.e.,...
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### Morita-equivalence of groupoids and classifying topoi: correct definition

The first comment to this post points out that, given two (topological or localic) groupoids, they can be non-Morita-equivalent evenif their classifying topoi (topoi of equivariant sheaves) are ...
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### Understanding the monadicity of groupoids over splittings

In the paper The shift functor and the comprehensive factorization for internal groupoids by Bourn, the author proves that for a fixed finitely complete category, the category of internal groupoids is ...
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### Inversion map of a Lie groupoid is a diffeomorphism

The inversion map $Inv:\mathcal G\to \mathcal G$ of a Lie groupoid $\mathcal G$ is given by $Inv(g)=g^{-1}$. And I want to show this inversion map is a diffeomorphism. Any suggestions will be ...
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### Does it make sense to examine properties of $(G,\circ)$ if we, in the meantime prove that $(G,\circ)$ is not a groupoid?

If, for example, we are given a set $G$ and an operation $\circ$ and we have to examine properties of that operation on that set (closeness, associativity, commutativity, exsistance of neutral, ...
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### Does the existence of only right neutral mean that there can only exist right inverse?

Let $(G,\circ$) be groupoid. If there exists only right neutral, does that mean that there can exist only right inverse? To put it in another way, does the existence of only right neutral mean that ...
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### Gleason Yamabe for groupoids

A colleague of mine seems convinced that there is a Gleason-Yamabe type theorem for locally compact groupoids. Does anyone know if this is true? If so, any references would be most appreciated.
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### Why is a groupoid required to be a small category?

I'm learning category theory and every definition of a groupoid has required that a groupoid be a small category (or some equivalent requirement) but I don't really know why. So what is the reason ...
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### Is the category of groupoids a Lawvere thory?

By which I mean a category of models for a Lawvere theory. I have not seen this anywhere, so I wonder if something goes wrong with this category.
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### Easily visualizable examples of Lie groupoids?

The goal is to understand the Poisson cohomology of Poisson manifolds, which according to the nCatLab, is "just" the Lie algebroid cohomology of the corresponding Poisson Lie algebroid. Chasing down ...
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### Representations of groupoid algebras

In reading through Khalkhali's Noncommutative Geometry text, I came across something I don't understand. Let $\mathfrak{G}$ be a discrete groupoid, and for each $x\in Obj(\frak{G})$, define the *-...
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### The difference between weak Kan complexes and Kan complexes

Let $X$ be a simplicial set and let $\wedge_i^2$ be the $i-th$ horn of the simplicial set $\Delta^2$. The Kan condition is the horn filling condition for $i=0,1,2$ and the weak Kan condition is the ...
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### Coseparating family in the category of groupoids

Does the category of (small) groupoids admit a small coseparating/cogenerating set of objects? I suppose $\mathbf{Cat}$ doesn't, but so far I have no clue about $\mathbf{Grpd}$.
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